Let O designate the centre of an equilateral triangle. Points U-Z are chosen at random within the triangle. We have learnt that points U,V,W are each nearer to a (possibly different) vertex than to O; while X,Y are each closer to O than to any of the vertices.

Show that triangle XYZ is more likely than triangle UVW to contain the point O within its interior.

Nice going, Steve and Charlie, for working through the UVW case so smoothly.

I am consistently surprised at my inability to predict the response to the problems I pose. This problem was moderately hard, and although I'm not surprised to see a (partial) solution, I didn't expect it so quickly.  

I recognise computer exploration as a useful tool in characterising an outcome. Charlie used it to convince himself the probability for case XYZ should be one quarter. Perhaps that information could be helpful to guide further efforts (I'm not sure how). More likely, it serves as a useful comparison for validating a solution. Under no circumstances though ought the experimentally generated result be confused with the solution. The problem remains unsolved, and despite the reflections above, I'm willing to risk the statement that the outstanding XYZ case is no harder than the UVW case.

I hope that Steve, Charlie and others may reach for glory by making the attempt.